Experimental and numerical investigation of the acoustic response of multi-slit Bunsen burners

Combustion and Flame 156 (2009) 1957–1970

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Combustion and Flame

journal homepage: www.elsevier .com/locate /combustflame

Experimental and numerical investigation of the acoustic response of multi-slitBunsen burners

V.N. Kornilov a,*, R. Rook b, J.H.M. ten Thije Boonkkamp b, L.P.H. de Goey a

a Department of Mechanical Engineering, Combustion Technology Group, TU/e, P.O. Box 513, 5600 MB Eindhoven, The Netherlandsb Department of Mathematics and Computer Science, Scientific Computing Group, TU/e, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

a r t i c l e i n f o

Article history:Received 15 January 2009Received in revised form 10 June 2009Accepted 15 July 2009Available online 6 August 2009

Keywords:Premixed Bunsen flamesThermo-acoustic transfer functionExperimental and numerical research

0010-2180/$ - see front matter � 2009 The Combustdoi:10.1016/j.combustflame.2009.07.017

* Corresponding author. Fax: +31 40 2433445.E-mail address: [email protected] (V.N. Kornilov).

a b s t r a c t

Experimental and numerical techniques to characterize the response of premixed methane–air flames toacoustic waves are discussed and applied to a multi-slit Bunsen burner. The steady flame shape, flamefront kinematics and flow field of acoustically exited flames, as well as the flame transfer function andmatrix are computed. The numerical results are compared with experiments. The influence of changesin the mean flow velocity, mixture equivalence ratio, slit width and distance between the slits on thetransfer function is studied, both numerically and experimentally. Good agreement is found whichindicates the suitability of both the experimental and numerical approach and shows the importanceof predicting the influence of the flow on the flame and vice versa. On the basis of the results obtained,the role and physical nature of convective flow structures, heat transfer between the flame and burnerplate and interaction between adjacent flames are discussed. Suggestions for analytical models of pre-mixed flame–acoustics interaction are formulated.

� 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1. Introduction

Acoustic instability of combustion processes is a phenomenonwhich can occur in practically any type of combustion devicewhere the flame (or, more generally, a source or sink of energy)is enclosed in a vessel. In some applications the pulsating combus-tion is specially arranged to achieve high performance of the com-bustion and heat-transfer processes and to minimize harmfulemissions. However, in most practical combustors, thermo-acousticinstability is an undesirable effect, which is often difficult to predictand eliminate.

A general explanation of thermo-acoustic instability was firstgiven by Rayleigh at the end of 19th century. The physical natureof the problem is related to the positive closed loop of interactionsbetween acoustic waves in the complete system (vessel) and theflame. Acoustic waves can lead to an increasing heat release rateof the flame, which can amplify the acoustic wave, leading to an in-crease of the acoustic energy in the system, etc., provided the well-known Rayleigh criterion is satisfied.

In spite of significant progress in the theory of thermo-acousticinstability, the problem of combustion-generated noise is still notwell understood, both on a fundamental level and in practicalapplications. A fundamental description of flame–acoustic wave

ion Institute. Published by Elsevier

interaction is still lacking, and consequently also proper designingguidelines for combustion devices, because there are many possi-ble causes for the feedback mechanism between flame and acous-tic waves. There exist many ways to arrange this flame–acousticwave coupling, depending on the specific flame and/or burner.Therefore, each type of flame requires a separate investigation.

In the present paper we will consider the response of fully pre-mixed laminar Bunsen-type flames on a multi-slit burner to a fluc-tuating velocity field, resembling an acoustic wave. For theseparticular flames two different phenomena of flame–acousticsinteraction can be expected a priori. First, small flames on themulti-slit burner can display an oscillating heat release rate dueto flame surface undulations, as is typical for Bunsen flames. Sec-ond, the heat loss rate to the burner deck near the flame foot canoscillate, resembling the case of a flat flame stabilized on a surfaceburner. These two phenomena are expected to be responsible for afluctuating heat release.

The choice of flames on a multi-slit burner as an object of inves-tigation is motivated by the following factors. First, from a practicalpoint of view, similar flames are widely used in the burners ofsmall and moderate scale combustors, like domestic and district-heating boilers, dryers, etc. Second, with regard to fundamentalresearch purposes, the slit configuration provides a close to two-dimensional geometry which is convenient for numerical simula-tion in Cartesian coordinates. Furthermore, the multitude of flamesallows applying symmetry boundary conditions, thus facilitatingthe numerical simulations significantly.

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1958 V.N. Kornilov et al. / Combustion and Flame 156 (2009) 1957–1970

The approach in this paper is to investigate the response of theflames to a fluctuating velocity field, both experimentally andnumerically. In the numerical simulations, a model is used whichincludes the Navier–Stokes and transport equations for a two-dimensional flame on a burner, in which the full interactionbetween flame, flow and burner is taken into account. The experi-mental research includes both global characterization of the flamethermo-acoustic behavior and detailed spatially and temporallyresolved measurements of the flame front. Real and numericalexperiments are conducted, some of which have identical burnergeometry, flow and excitation parameters. In this way we candirectly compare experimental and numerical results of the flamefront motion and flow field. Furthermore, the measured andnumerically computed thermo-acoustic transfer function (TF) andtransfer matrix (TM), which characterize the flame as a lumpedelement, are compared and analyzed.

In the present contribution a numerical model, which providesthe possibility to reproduce experimental results, will be devel-oped and validated. Next, using the synergy of both numericaland experimental results, some insight in the physics of thermo-acoustic behavior will be given.

The paper is organized as follows: In Section 2, a genesis and thecurrent state of knowledge of thermo-acoustic behavior of laminarBunsen-type flames is presented. The experimental configurationis reported in the next section, including an outline of the experi-ments performed. In Section 4, the model used in our numericalsimulations together with the numerical data post-processing pro-cedures are described. The core of the paper consists of a compar-ison between experimental and numerical results of the flamethermo-acoustics, and will be presented in Section 5. This contri-bution ends with a discussion of the results obtained followed byconclusions.

2. Thermo-acoustics of premixed flames: survey of existingmodels

Recent reviews of different aspects of flame–acoustics interac-tion, including Bunsen-type flames, were presented by Candel [1]and Lieuwen [2]. Below we will attempt to trace how progress inthe understanding of the physical nature of this interaction was

Fig. 1. Experimental flame TF gain (a) and phase (b); polar plot (c); real and imU ¼ 0:9; V ¼ 150 cm=s and d ¼ 1:0 cm) and a flat flame stabilized on a burner with per

incorporated into several models. On the basis of this analysis to-gether with results of the present study, we will suggest possibleways to improve both models and experiments.

2.1. Approaches to characterize flame thermo-acoustics

The three most widely used methods to characterize the inter-action between flames and acoustic waves in the linear limit arebased on, (i) the thermo-acoustic transfer function, (ii) the acoustictransfer matrix and (iii) the pyro-acoustic amplification factor.

The main part of our discussion will be based on the transferfunction concept. It means that the response of a flame to acousticwaves is characterized in the frequency domain by the so-calledflame thermo-acoustic transfer function, which is defined as theratio of the relative heat release rate perturbation (response) tothe relative flow velocity perturbation (stimulus), i.e.,

TFðf Þ :¼ q0=�qu0=�u

; ð1Þ

where q is the heat release, u the velocity and f the frequency of theperturbation. A prime ð0Þ denotes the perturbation and a bar ð�Þ theaverage value of a variable. If the response q0 at a forcing frequency fis linearly proportional to the perturbation u0, then the TF is inde-pendent of the amplitude of the perturbation. This linear regimewill be considered in the present paper. TFðf Þ is a complex numberand can be characterized either by its magnitude (gain) and phase(Fig. 1a and b), or by its real and imaginary parts (Fig. 1d).

The concept to characterize the thermo-acoustics of a flame viaan acoustic transfer matrix is a purely acoustical approach whereparts of an acoustic system (vessel) are considered as lumpedacoustic elements, so-called two-ports. These elements relate theinput vector of acoustic variables, the acoustic pressure and veloc-ity ðp0up;u

0upÞ to the output vector ðp0down; u

0downÞ. The TM approach is

most suitable to describe longitudinal acoustic waves in the linearregime of oscillations. The acoustic elements are represented bythe so-called transfer matrix, which provides the relation betweeninput and output, i.e.,

p0down

u0down

� �¼Mðf Þ

p0up

u0up

!; Mðf Þ ¼

Mppðf Þ Mpuðf ÞMupðf Þ Muuðf Þ

� �: ð2Þ

aginary parts (d) of the TF for a single Bunsen flame on a tube (lines 1 forforated brass deck (lines 2 for U ¼ 0:9; V ¼ 10 cm=s and T ¼ 260 �C).

V.N. Kornilov et al. / Combustion and Flame 156 (2009) 1957–1970 1959

It is possible to show (see e.g. [3–6]), that if the flame/burner isan acoustically compact element, then, in the limit of low-Machnumbers, only the element Muuðf Þ is sensitive to the presence ofa flame (heat release). This element relates the oscillating (acous-tic) components of the velocity in the downstream (burnt) and up-stream (unburnt) parts of the flame/burner, where the acousticfields are assumed to be planar waves. The element Muuðf Þ is acomplex number with magnitude equal to u0down=u0up

�� and phaseequal to the phase difference between u0down and u0up.

The approach of flame pyro-acoustic amplification is presentedin [7] and will be not used here. A detailed comparison betweenthe different methods is the subject of a separate investigation;however, some of the aspects will be addressed below. A relationbetween the TF and TM approaches is discussed in [3,5].

2.2. Experimental evidence of thermo-acoustics of laminar premixedflames

The acoustic response of Bunsen-type flames has been studiedintensively over the years. Putnam’s experiments [8] with manydifferent systems in which a self-sustained acoustic instabilitycan be observed, forced him to conclude that there is a combustionprocess time lag and that this time lag is equal to a traveling timeof gas particles from the burner outlet to the mean position in theflame zone. By nature this time lag is a ‘system time delay’, in otherwords, the flame effectively responds to a flow perturbation sometime after it is applied at the burner outlet. This observation hasbeen verified in many experimental studies afterwards. Amongthese early measurements the experiments of Matsui [7], Sugimotoand Matsui [9] and Goldschmidt et al. [10] are representative. Inthe last decade the TF of several Bunsen-type premixed flameswas intensively studied, such as conical, inverted conical (V-form)and M-form flames, as well as flames anchored at multiple perfo-ration burner decks; see [11–15].

Summarizing the cumulative experimental material of differentBunsen-type flames the following qualitative conclusions can bedrawn. Besides the time delay property of the Bunsen-type flameTF, which was observed by practically all researchers, there areseveral typical features of the flame TF which also require physicalinterpretation; see Fig. 1. First of all, the TF gain shows a globaldecay when the perturbation frequency increases. This ‘low-passbehavior’ could be anticipated a priori; however, the decay behav-ior is non-trivial. The gain of the TF has distinct minima and max-ima. Furthermore, the presence of a weakly frequency dependentcomponent of the TF can also be recognized. It is evident fromthe high frequency saturation followed by a slow drift of the TFphase and gain. Another feature, typical for V-form, M-form andmultiple flame configurations but absent for a free conical flame,is the overshoot of the TF gain above 1 for some excitationfrequency.

Regarding the thermo-acoustic TF of flat burner surface stabi-lized flames the following main features can be observed; seeFig. 1. The flame TF exhibits a low-pass filter behavior, shows a lim-ited phase change /ðf Þ, asymptotically approaching p at high fre-quencies, and displays a large gain Gðf Þ at low frequencies(resonance, [6]).

2.3. The genesis of thermo-acoustic models of premixed Bunsen-typeflames

A first attempt to build a theory of the response of conicalflames to an acoustic flow excitation was developed by Merk[16] on a semi-phenomenological basis. The proposed model leadsto a first-order equation for the heat release rate and accordingly tolow-pass filter features of the flame response to excitation. For thecharacteristic time of the flame response Merk proposed to use 1/3

of the ratio of the flame height H to the flow velocity V at the flamecone base, the so-called ‘convective time of the flame cone’. Notonly the physical background of this estimation is obscure but, fur-thermore, the physical nature of this time is a system relaxationtime and not a system response time lag. A manifest discrepancybetween experiment and theory is that for the high frequency limitMerk’s theory predicts a phase for the transfer function which isasymptotically approaching p=2, while Putnam’s experiments gaveevidence to a much larger phase.

The next stage in modeling was based on the observations ofperturbed flame form motion. It was reported by Markstein [17]and Blackshear [18] that flame front undulations periodically raisethe flame cone anchoring zone and travel towards the flame tipwith a velocity close to the convective flow velocity. Accordingly,theories developed in the 1970s prescribe the flame front pertur-bation as convective waves on the surface of the flame cone. Theflame heat release rate perturbation can be calculated assumingthat the flame surface area variation is the main cause of the heatrelease oscillation.

Different variants of these models were reviewed, further devel-oped and presented in a clear way by Matsui [7,9]. The results ofthese models suffer from the same shortcoming as Merk’s model.The models predict a low-pass filter behavior of the flame TF. Atthe same time, direct measurements of the TF by Sugimoto andMatsui [9] indicate that the gain of the TF has a complicated jaggedform with multiple minima and maxima while the TF phase showsan almost linear increase with the perturbation frequency, typicalfor systems with a time delay. To eliminate this grave discrepancybetween theory and experiment Matsui and Sugimoto proposed amodification of the model by including an exponential growth ofthe flame surface perturbation amplitude as it propagates alongthe flame cone. They have affirmed their idea by measurements;however, the conclusion of an increasing perturbation amplitudewas an artifact of their data processing procedure [14,19] andwas not confirmed by later research.

A new stage in modeling was initiated by the idea to describethe kinematics of the excited flame front by the so-called G-equa-tion [20]. The fundamentally new idea in this approach is that theflame front perturbations are not prescribed a priori, but naturallyarise as the flame front responds to an applied flow perturbation.Therefore, in the framework of kinematic models the input is theoscillating flow field and the output is the flame surface area oscil-lation, which is directly proportional to the heat release rate,assuming that the normal flame speed is constant over the flamesurface. In early versions of the kinematic models, the flow pertur-bations were prescribed as bulk streamwise oscillations [21]. Thecomplete theory is presented in [22]. The model predicts theformation and further convection of flame front undulations whichwas prescribed in previous models and observed experimentally.However, the resulting TF once again displays a low-pass filterbehavior and a phase saturation at p=2.

The reason why convection of the flame cone undulations doesnot lead to a convective time delay behavior of the TF is hidden inthe nature of the surface area calculation; for more details see [14].The conclusion which is important is that convection of flame frontperturbations is not sufficient to explain the time delay property ofthe flame cone area TF.

Several ideas were proposed to overcome the discrepancybetween the results of modeling and experiments, see the reviewin [14]. Currently, the most plausible explanation is that the flowfield at the burner outlet creates periodical perturbations of aconvective nature superimposed on the acoustic wave. These per-turbations resemble the peristaltic motion of a jet and can bedescribed as convective traveling waves. The correspondingvelocity field inside the flame cone was observed and characterizedexperimentally [23–25]. It was shown that the measured


oscillating velocity field as input to the kinematic model results ina TF which exhibits the proper time delay behavior close to exper-imental results [26]. Simplifying the velocity field even allows toderive an analytical expression for the flame TF [22], having thesame time delay features. Furthermore, it was demonstratedexperimentally, that an excited flame front without convectivewaves yields a TF which does not display the observed time delayproperty [25].

It is interesting to note that the parameter values and the qual-itative behavior of the convective waves in the flow upstream ofthe flame cone differ from the waves of the well-known convectivestructures at the outlet of an excited cold jet, referred to as vortexshedding [24,27]. Currently, it is not clear what causes the differ-ence. It can be either the mean effect of the pressure change dueto gas expansion (acceleration) at the flame front, or the effect ofthe time-dependent (oscillating) flame retroaction to the upstreamflow. Regarding the last possibility, two hypotheses are formulated.The first one concerns the effect of the curved flame front on theflow upstream of the flame [24]. The second hypothesis considersthe radial motion (in and out) of the anchoring point as the maindriver of the peristaltic jet motion [25].

It was shown [14] that the nonmonotone form of the TF gain isprobably a direct consequence of the fact that the TF of Bunsen-type flames contains in addition to a purely kinematic part anadditive component, which is relatively small and has a weakerfrequency dependence; see the convergence point of the TF spiralin Fig. 1c. This TF ‘offset’ also shows up at saturation followed byslow drift of the TF gain and phase when the excitation frequencyincreases. There is evidence that the TF offset originates from theflame foot region and is probably associated with the motion ofthe flame anchoring point [14]. In this zone the role of heat transferbetween the flame and burner surface plays a dominant role andtherefore we conjecture that the TF offset resembles the TFdescribing the interaction of a flat surface burner-stabilized flamewith acoustic waves. There is only one, first attempt to build amodel which combines both effects of flame cone kinematics andoscillating heat loss to the burner surface [28].

The TF gain overshoot above 1 for a certain frequency range wastheoretically described in the framework of kinematic models forV-flames [22], which is in accordance with experimental observa-tions. The cause of the amplification of the response over its quasi-steady value lies in the interference between perturbations whichoriginate from the anchoring point and travel along the flame, anda convective perturbation of the flow pattern [29]. A TF gain above

Fig. 2. Burner setup (a) and

1 was also measured for multiple flame burner heads [15,30]. Atthe same time, both the theoretical and the experimental TF of asingle conical flame shows a TF gain less than 1 for the whole fre-quency range.

Until very recently the thermo-acoustic behavior of a configura-tion of multiple identical flames has been interpreted on the basisof single (individual) flame behavior. The qualitative features of asingle flame and identical multiple flames are similar. In bothcases, the gain of the TF as a function of frequency has a compli-cated jagged form and the TF phase has a time delay behavior. Inspite of the qualitative resemblance the two flame configurationsare not identical. At least, one more burner parameter, viz. the in-ter-flame spacing (pitch) becomes important for the TF of a multi-ple flame burner [15]. The overshoot above 1 of the TF gain ofmultiple flames is an extra evidence of the qualitative differencebetween the single and multiple flame configurations. To ourknowledge there are no models of flame thermo-acoustics that in-clude flame-to-flame interaction.

Summarizing the progress in understanding, we can concludethat thermo-acoustic models for premixed Bunsen-type flamesevolved from the prescription of the flame surface area dynamicsin the early models, towards the prescription of the flame formperturbation in the theories in the period 1970–1980 to theprescription of the oscillating flow field in modern theories. A nextlogical step would be to predict the flow dynamics starting fromthe prescribed plane acoustic field at some position upstream ofthe burner. This step will probably require the incorporation ofthe flame retroaction on the upstream flow. Other necessary exten-sions of the models should include (i) intrinsic mechanisms offlame anchoring to the burner (or flame-holder) which accountsfor heat transfer between flame and burner surface and (ii) mutualinteraction between adjacent flames, probably via the downstreampart of the flow. The aim of the present research is to provideresults which can serve as basis for future analytical investiga-tion/models.

3. Experimental method

3.1. Experimental setup

The burner which was used in the present study consists of avessel with a flat perforated disk inserted on top of it; see Fig. 2a.The disk contains a series of 12 mm long rectangular slits, eachof width d, whereas l is the distance between adjacent slits. The

calculation domain (b).


pitch then equals lþ d and the porosity n ¼ d=ðdþ lÞ. The slits areperforated in a steel disk of 0.5 mm thickness. A mixture of meth-ane and air at ambient conditions ðp ¼ 1:0 atm and T ¼ 293 KÞ,with an equivalence ratio U and a velocity �uu (approaching thebulk velocity below the plate, leading to an average velocity V inthe slit) is used to stabilize the steady flames. The burning velocityof the mixture is sL. The burner plate temperature was measuredby a K-type thermocouple embedded in the center of the burnerplate. The temperature reaches 100—150� C due to steady combus-tion. The gas flows are controlled with mass flow controllers (MFC)installed far enough from the burner to allow a perfect mixing andto avoid a possible acoustic influence on U and/or �uu. To impose aflow velocity perturbation u0, a loudspeaker operated by a puretone generator was installed upstream in the mixture supply tube.Because the mixture composition is unperturbed and ambient airentrainment cannot be significant it is known that to measurethe flame heat release rate, the chemiluminescence intensity ofOH� can be used as an appropriate indicator (see e.g. [31] and fur-ther references). To monitor the velocity oscillation a hot-wire ane-mometer was installed 10 mm upstream, just beneath a slit in theburner vessel (see Fig. 2a).

Flame front kinematics and flow field visualizations wereperformed via the direct phase-locked video-recording of the flamenatural emission (a CCD camera, Kodak Megaplus ES1.0, was used)and flow velocity field reconstruction was done by the conven-tional Particle Image Velocimetry (PIV) technique. The flow wasseeded with Al2O3 particles with average size of approximately1 lm. Both the upstream and flame region of the flow were illumi-nated by a light sheet of an Nd:YAG pulsed laser. For this purposethe upper part of the burner vessel is made optically transparent.Care was taken to minimize the light reflections on the entranceand exit windows of the laser sheet which were made in the upperupstream burner section. The processing of raw PIV image pairswas conducted by the PIVview2C software.

3.2. Procedure of TF data processing

The experimental TF is found by measuring the flame responseat a large number of different frequencies. According to definition(1) of the thermo-acoustic TF, its amplitude (gain) and argument(phase) can be reconstructed from the measured time series ofthe relative flow velocity perturbation u0=�u and the relative heatrelease rate perturbation q0=�q. Raw experimental data consist of0.5 s samples of u0ðtÞ and I0OH� ðtÞ time histories digitized with asampling rate of 20 kHz. A perturbation amplitude u0=�u of approx-imately 5–10% was chosen, for which the flame response can beconsidered linear [14]. The gain of the TF was calculated as theratio of the amplitudes of the Fourier transform of I0OH� ðtÞ and theFourier transform of u0ðtÞ. The phase difference between both sig-nals was determined by a cross-correlation analysis. The measuredvalues can be presented either in the form of a frequency depen-dent gain Gðf Þ and phase delay /ðf Þ (see Fig. 1a and b) or in a polarplot where Gðf Þ represents the radial length and /ðf Þ the angle (seeFig. 1c).

4. Numerical modeling

The code LAMFLA2D [6,32] is used to simulate the response oflean methane–air flames to velocity perturbations. The code solvesthe primitive variable formulation of the conservation laws fortwo-dimensional, low-Mach number reacting flow. It is based ona one-step chemical reaction model for the species CH4; O2; CO2;

H2O and N2. LAMFLA2D uses the following numerical methods: asecond order finite volume/complete flux scheme for space discret-isation, the implicit Euler method for time integration, a pressure-

correction method to decouple the pressure computation and anonlinear multigrid method and GMRES to solve the discretizedsystem. For more details see [6,32,33].

The most important physical and chemical models used are pre-sented in the following. The diffusion fluxes are modeled using aFick-like expression with the mixture-averaged diffusion coeffi-cients Di;m given by

Di;m ¼ ð1� YiÞX

j–iXj=Dij

.; ð3Þ

where Xi and Yi are the mole and mass fractions, respectively, andDij are the binary diffusion coefficients [34]. The transport equationsfor the species CH4; O2; CO2 and H2O are solved. The mass fractionof the Nth, abundant, species N2 is computed from the constraintPN

i¼1Yi ¼ 1 to assure that the sum of all diffusion fluxes equals 0.A semi-empirical formulation is applied for the conductivity, i.e.,

k ¼ 12

XN

i¼1

Xiki þXN

i¼1

Xi=ki

!�10@

1A; ð4Þ

where ki is the thermal conductivity of the ith species. The transportcoefficients Dij and ki are tabulated in terms of polynomial coeffi-cients, similar as in the CHEMKIN package [35]. The thermodynamicproperties are also tabulated in polynomial form [36].

We apply the following one-step overall irreversible reactionmechanism in the numerical study:

CH4 þ 2O2 ! 2H2Oþ CO2 ð5Þ

with the reaction rate of methane given by [37]:

_qCH4 ¼ �AqmþnYmCH4

YnO2

expð�Ea=RTÞ: ð6Þ

The reaction parameters were fit to experiments to predict thecorrect relation between the burning velocity and flame tempera-ture, for flat adiabatic and burner-stabilized flames, in the range0:8 6 U 6 1:2 and optimized for U ¼ 0:8 [38], resulting in m ¼2:8; n ¼ 1:2; Ea ¼ 138 kJ=mol and A¼2:87�1015ðkg=m3Þ1�m�n s�1.The effect of heat losses is incorporated in the fitting procedureto make sure that the flame stabilizes due to heat transfer to theburner. Accordingly, we can surmise that the phenomenon of flameanchoring, i.e., the stand-off distance, is accurately modeled. It wasshown in earlier studies [38] that this model accurately describesthe global behavior of steady burner-stabilized flames. In [39] itwas shown that this mechanism is also well-suited to model theresponse of one-dimensional lean methane–air flames to low-fre-quency acoustic perturbations. In the current paper we restrictourselves to premixed methane–air flames with U¼0:8.

Only a small part of the repetitive flame structure is computedon a domain of width half the pitch ðlþ dÞ=2, using symmetryboundary conditions at both sides; see Fig. 2b. The inflow belowthe burner plate is also taken into account in the simulations usinga flat velocity profile with perturbation, i.e., u ¼ �uþ u0 as inflowcondition. This condition accurately models the approachingacoustic wave at the inflow, since acoustic waves have infinitewave length and pressure fluctuations are not needed in the limitof the Mach number Ma! 0. The outflow is simply modeled bysetting to 0 the normal derivatives of the velocity and the combus-tion variables, which is accurate if the outflow boundary is suffi-ciently far away from the flame front. The temperature ofdifferent burner decks was not computed but was set at 100 �C.The influence of the burner deck temperature and thickness wasnot studied here. However, as follows from the results in [13],the influence of the thickness of the burner plate on the burnerthermo-acoustics is weak and becomes noticeable when it is sig-nificantly larger than the perforation size.


4.1. Post-processing of numerical data

Most flame TF and TM are computed during a single computa-tion, i.e., by modeling the flame response to a small but instanta-neous velocity change at the inlet. This perturbation contains theresponse information of all frequencies which can be found by Fou-rier analysis. With respect to calculation time, applying broad bandexcitation (like a jump of the inlet velocity) is much more effectivethan the calculation of the flame response for each pure tone sep-arately. However, the requirements on the response linearity aremuch more restrictive in the case of broad band excitation thanin the case of single frequency response. The perturbations shouldbe small enough to avoid nonlinear flame response which can in-duce spurious components of the response at higher harmonics.Application of low magnitude perturbations causes the problemof signal-to-numerical noise ratio. The detailed analysis of thisquestion can be found in [40]. The correctness of the flame TFreconstruction from the response to a stepwise perturbation ischecked by varying the velocity perturbation magnitude. The timeseries of the flame heat release rate was calculated via the integra-tion over the entire calculation domain.

The time history of inlet and outlet velocities required for theburner TM calculation were determined in the correspondingzones of the calculation domain, when a stepwise excitation wasapplied. We checked that almost plane distributions of mean andoscillating velocities were retrieved in the far downstream partof the calculation domain.

Flame front kinematics and flow pattern features under the ac-tion of pure tone excitation were visualized during both numericaland real time (phase) and spatially resolved experiments. Innumerical experiments, five periods of oscillation were calculatedwhich was sufficient to reach the steady-state periodic responseof the flame and flow. As initial condition the corresponding steadyflame was used. To minimize the transition time to the steady-state oscillation a smoothly growing amplitude of the tone pertur-bation was imposed.

Besides the heat release transfer function TF, we introduce thetransfer function TFA, which relates the relative response of theflame surface area A (defined in several different ways, see below)to the velocity perturbation, i.e.

TFAðf Þ :¼ A0=Au0=�u

: ð7Þ

Flame front kinematics and the calculation of TFA supposes theflame front to be a line (in two dimensions) or a surface (in threedimensions) which separates burnt and unburnt gases. Accord-ingly, a procedure to determine this line in the reacting flow fieldshould be specified. For the case of Bunsen-type flames it is naturalto take for the position of the flame front the line where the valueof the heat release rate distribution has its maximum. However,the flame front is not attached to the burner surface and a locationfor the flame end (foot) should be specified as well. When theflame front and flame end point are reconstructed, the flame sur-face area can be determined as the length of the flame line fromits tip to the end point.

The following criterion was used for the flame anchoring point:it is the point where the heat release rate, as a function of the coor-dinate along the flame front, has the maximal rate of change. Thephysical meaning of such a definition is the following: the endpoint is the location where the reaction rate along the flame fronthas its maximal gradient of decay. A similar procedure of flamefront tracking was used before [14,19] for the analysis of experimen-tal data in the case of much larger Bunsen flames. However, for theslit burner configuration considered here the noise-to-signal ratioof experimental images of perturbed flames is not sufficient to

obtain the accuracy of the flame front kinematics reconstructionwhich is necessary to compute the transfer function. Therefore, onlythe results of numerical simulations will be post-processed. Further-more, the accuracy of numerical simulations was not sufficient tocalculate the flame surface area TFA from the numerical experimentwith a stepwise excitation of the inlet velocity. Because of this, thecalculation of the flame response to a pure tone excitation in a lim-ited range of frequencies was used to calculate the flame areaoscillation.

Besides the physically reasonable definition described above,some other, easier, ways to define the flame anchoring point werealso examined. Two other methods are tested: (i) the flame is trun-cated by the vertical lines emanating from the slit edge, which isequivalent to the artificial ‘attachment’ to the slit edges and (ii)the flame front was reconstructed from the 1200 K) isotherm(see the line in Fig. 10c). This temperature was arbitrarily selectedas one where typically the reaction layer of the methane–air flamestarts.

5. Results

Two classes of experimental and/or numerical results will bepresented. The first one concerns measurements of the flame TFfor a set of burner, flow and excitation parameters. This is a globalflame analysis which allows us to assess the experimental andnumerical methods to determine the flame TF. Some hypothesesabout the physical mechanisms of the flame response will be pro-posed and examined on the basis of the TF. Moreover, the numer-ically computed TF and TM will be compared and the relationbetween them will be discussed. The second class deals with adetailed comparison between steady and acoustically exitedflames, i.e., the flame motion and the flow are resolved temporallyand spatially. The resulting data will lead to a better understandingof the thermo-acoustic behavior of flames.

5.1. Comparison of experimental and simulated flame TF and TM

First, the measured flame TF is analyzed and compared withnumerical results for a flame with U ¼ 0:8; V ¼ 100 cm=s; d ¼2:0 mm and l ¼ 3:0 mm; the so-called representative case. Next, aparameter study is presented for the TF with varying U; V ; dand l. Moreover, the measured and computed shape of the steadyflame is presented, in order to facilitate further analysis of theTF. The TM was only determined via numerical simulations.

5.1.1. Flame TF for the representative caseFig. 3 compares the steady flame shapes (c) and TF (gain Gðf Þ (a)

and phase /ðf Þ (b)) for the representative case. The experimentalpart shows a chemiluminescence photograph while the chemicalsource term is displayed in the modeling results of Fig. 3c. The cor-respondence is reasonable despite that different quantities arevisualized. The experimental flame has a height H of approximately4.5–4.7 mm while the numerically computed flame is slightlysmaller with a height of 4.5 mm. Note also that individual flamesstabilize on the slits and do not merge near the foot in both theexperiment and the numerical simulations.

The experimental and numerical flame TFs are compared inFig. 3a and b. The correspondence of gain and phase is reasonable.All qualitative features of the experimental TF are captured bynumerical simulation. The time delay property (almost linear in-crease of the TF phase with frequency) is accurately predicted.The effect of the TF gain overshoot of the quasi-steady responseis also accurately computed.

The TF phase saturation property is somewhat overpredicted,which is reflected by the fact that the numerical TF phase

Fig. 3. Comparison of experimental and numerical flame TF gain (a) and phase (b) for the representative case, including a comparison of the steady numerical andexperimental flame structure (c). Parameter values are: U ¼ 0:8; V ¼ 100 cm=s; d ¼ 2:0 mm and l ¼ 3:0 mm.


approaches a constant phase of approximately 11 rad forf > 450 Hz, while the experimental TF phase is still increasing. An-other consequence of the TF offset overprediction is the wigglybehavior of the computed TF gain compared to the experimentalone. The difficulty to predict the TF offset accurately is due to itshigh sensitivity to small inaccuracies of the flame (mixture)parameters, both experimentally and numerically.

5.1.2. Effects of U; V ; d and l on the TFFig. 4 shows measurements and simulation results of the flame

TF for varying V while U ¼ 0:8; d ¼ 2:0 mm and l ¼ 3:0 mm. Thefigure indicates that the numerical simulation is very well capableto predict the behavior of the experimental results for all cases. TheTFs for the lowest velocities look very similar to those of a flatburner-stabilized flame (low phase for all frequencies and high

Fig. 4. Comparison of experimental flame TFs (a and b) with computed TFs (c and d) for vU ¼ 0:8; d ¼ 2:0 mm and l ¼ 3:0 mm; TF gain (top a and c), TF phase (bottom b and d).

gain at small frequencies; see lines 2 in Fig. 1), since the individualflames are very small and the major part of the combustible mix-ture is consumed as in a flat surface burner. For higher velocitiesthe flame height increases and much more of the mixture is con-sumed by the longer Bunsen-type flames, and as a result, the con-tribution of the Bunsen flame TF to the multi-slit TF increases.Therefore, the phase becomes very similar to the Bunsen-typeflame behavior, with a constant slope at low frequencies and satu-ration at a constant phase at higher frequencies (see lines 1 inFig. 1). The slope of the phase, proportional to the convective times :¼ H=V , does not change when V is varied, because the laminarflame height H also increases for increasing velocity. The saturationlevel increases with V like in the case of a single Bunsen flame dueto the shift of the cutoff frequency of the low-pass filter to higherfrequencies. A similar effect is seen in the gain: at low velocities,

arying velocity, i.e., V ¼ 50;62:5;75;100;125;150 cm=s. Other parameter values are:

Fig. 6. Comparison of measured flame TF (thick lines) with computed TF (thin lines)for varying l with U ¼ 0:8; V ¼ 100 cm=s and d ¼ 2:0 mm; TF gain (a), TF phase (b).


Gðf Þ looks like the surface burner gain, while the influence of theBunsen-type flame gain becomes more important for increasingV. This leads to the observation that the change in gain Gðf Þ is morepronounced than in the case of a single Bunsen flame [14]. The fre-quency range where the TF gain exceeds 1 enlarges with increasingflow velocity.

Fig. 5 shows measurement data and numerical results of theflame TF for varying slit width d and distance l with l=d ¼ 1:5,implying that the porosity is constant, while U ¼ 0:8 andV ¼ 100 cm=s. In this experiment the effect of purely geometricalscaling is revealed. The flame height H increases with increasingd while V remains constant so that the slope of /ðf Þ, proportionalto s, increases. The frequency range where the TF gain exceeds 1is larger for smaller flames. The correspondence between experi-mental data and numerical results is again excellent.

Fig. 6 shows measurements and numerical results of the flameTF for varying distance between the slits l while U ¼ 0:8; V ¼100 cm=s and d ¼ 2:0 mm. Increasing the distance between theslits leads to a separation between the attachment points of indi-vidual flames, a higher flame foot position and a larger flameheight. This, once more explains the increasing slope in the phaseplots, whereas the gain is hardly influenced. The frequency rangewhere the TF gain overshoots 1 is almost the same for all valuesof l.

Finally, Fig. 7 shows measurements of the flame TF for varyingU while V ¼ 100 cm=s; d ¼ 2:0 mm and l ¼ 3:0 mm. No compari-son with numerical results are given here, since we only didnumerical simulations for U ¼ 0:8. The slope of the phase /ðf Þ isproportional to s, which decreases for increasing U because theflame length H becomes smaller. A smaller flame is a result of ahigher burning velocity corresponding to a richer mixture. The gainis hardly influenced by changes in U as for the case of a single Bun-sen flame [14]. The region where the TF exceeds 1 decreases withincreasing U for the burner and flow parameters used in Fig. 7.

5.1.3. Numerical simulation of TM versus TFThe calculation of the element Muu of the TM was included in

the numerical simulation of the TFs presented in Fig. 4c and d forvarious velocities. The magnitude and phase of Muu are presentedin Fig. 8c and d by solid lines together with the correspondingTFs, presented in Fig. 8a and b. Note that the scale of the TF gain

Fig. 5. Comparison of measured flame TF (thick lines) with computed TF (thin lines) forexperimental and numerical steady flame structure (c).

is now linear. The quasi-stationary limit f ! 0 corresponds to theratio of the densities of unburnt and burnt gases which is the sameas the reciprocal ratio of the corresponding temperatures. In thehigh frequency range the magnitude of Muu approaches 1 and thephase tends to the 0 modulo 2p. This means that the flame doesnot alter acoustic waves of sufficiently high frequency. For all linesin Fig. 8c, except the line for V ¼ 150 cm=s, the frequency rangewhere the magnitudes of Muu exceeds the quasi-stationaryresponse level is slightly narrower than the corresponding rangeof the TF gain (see Fig. 8a). For high inlet velocity ðV ¼ 150 cm=sÞa secondary maximum of the magnitude of Muu in the frequencyrange 200–300 Hz can be recognized.

It is interesting to compare the phases of the TF and TM, seeFig. 8b and d. For the smallest flame ðV ¼ 50 cm=sÞ the phase ofthe TF saturates to a level slightly smaller than p whereas thephase of the TM decreases to 0. For other inlet velocities the abso-lute value of the TF phase typically saturates to a level between

varying d and l with l=d ¼ 1:5; U ¼ 0:8 and V ¼ 100 cm=s; TF gain (a), TF phase (b),

Fig. 7. Measured flame TF for varying equivalence ratio U with V ¼ 100 cm=s; d ¼2:0 mm and l ¼ 3:0 mm; TF gain (a), TF phase (b).


p=2 and p, modulo 2p, whereas the corresponding phase of Muu

tends to 0 modulo 2p. However, the slopes of the TF and TM phasesin the linear regime are the same, which implies the same time lag.

Based on the Rankine–Hugoniot jump conditions, a relationbetween the flame heat release TF and the element Muu of theacoustic TM can be established; see Eq. (8). The dotted linesdepicted in Fig. 8c and d are obtained using (8). The temperatureratios Tburnt=Tunburnt which are required to link Muu with the corre-sponding TF are calculated within the same numerical simulations.The temperature ratio slightly increases as function of the imposedflow velocity from 6.39 for the case of V ¼ 50 cm=s till 6.56 forV ¼ 150 cm=s.

5.2. Oscillating flow field and flame front kinematics for therepresentative case

Only the representative case is examined for a few frequenciesand amplitudes of perturbation. As the qualitative conclusions for

Fig. 8. Comparison of simulated flame TF (a and b) an

the different flames were the same, we will restrict ourselves inthis section to a steady flame and a flame exited at 200 Hz. Allother parameters are the same as for the representative case.

5.2.1. Steady-state flameFig. 9 presents the raw PIV image (a) and processed results in

the form of a vector field overlaid by a gray-scales plot of thestreamwise (b) and transversal (c) components of the measuredvelocity field, for the steady flame. Parts of the flow inside the bur-ner plate, as well as approximately 0.5 mm above and 1 mmbeneath the deck cannot be visualized because of difficulties dueto laser light and image reflections on the surface of the plate.Fig. 9d represents the computed and measured vertical (stream-wise) component of the velocity along the slit centerline, markedby ‘y’ in Fig. 9b and d. The measurement domain in streamwisedirection is smaller than the computational domain. Therefore, line1 in Fig. 9d is shorter than line 2. Because the experimental config-uration is not entirely two-dimensional and some flow contractionand expansion occurs in the direction perpendicular to the mea-sured section plane, the measured inflow (upstream) and outflow(downstream) velocities are smaller than the computed velocities.However, the flow velocity in the vicinity of the slit plate andinside the flame cone is accurately predicted.

As can be expected, contraction of the flow beneath the slitleads to a gradual increase of the vertical velocity and a horizontalcomponent in the deck upstream part of the flow. Inside the flamecone the velocity is almost constant, weakly decreasing (circa 20%)towards the flame front, followed by a rapid acceleration due to gasexpansion. The velocity in the flame tip increases roughly by afactor 2, which is much less then the temperature ratio: Tburnt=

Tunburnt � 6. At the same time the total flow acceleration (ratio offar downstream to far upstream velocities) corresponds well tothe gas expansion factor due to the temperature rise.

After the flame zone the flow quickly recovers an almost flatvelocity profile, see upper line in Fig. 9c, where the three lines rep-resent the distributions of the streamwise velocity along three hor-izontal planes (parallel to the burner plate). The correspondingcomputed velocity field approximates the measured velocity pro-files quite well (not shown in the figure).

Fig. 10a and b presents a comparison of the chemiluminescenceimage of the stationary flame (Fig. 10a) and the measured flowfield overlaid with the contour plot of the flow dilatation rate(Fig. 10b). The difference between the position of maximal

d TM (c and d) for the cases presented in Fig. 4.

Fig. 9. Measured flow field of the stationary flame; raw seeded flow image (a), PIV restored streamwise (b) and transversal (c) component of the velocity; comparison ofmeasured (line 1) and computed (line 2) distribution of streamwise velocity along the slit centerline (d). Parameter values are: U ¼ 0:8; V ¼ 100 cm=s; d ¼ 2:0 mm andl ¼ 3:0 mm.


dilatation and heat release rate is of the order of the flame thick-ness, which is of the order of the measurement accuracy. Accord-ingly, both quantities are equally suitable as indicator for theflame front position.

In the flame foot region a gas recirculation zone can be ob-served. Fig. 10g shows a close view of the PIV velocity field nearthe flame anchoring point. Numerical simulation allows resolving

Fig. 10. Comparison of measured (a, b, and g) and computed (c–f and h) results for therate; (c) heat release rate and isotherm of 1200 K; (d) velocity vectors overlaid with tempanchoring zone. Parameter values are: U ¼ 0:8; V ¼ 100 cm=s; d ¼ 2:0 mm and l ¼ 3:0 m

this flow phenomenon close to the burner deck in great detail;see Fig. 10h. The layer of rapid temperature increase, see Fig. 10dand e, is much thicker here, which is due to a lower heat releaserate (see Fig. 10c) and hot gas recirculation. The pressure field(Fig. 10f) is also depicted for the sake of completeness.

From the analysis above we conclude that the spatial structureof the steady flame matches the experiment well. Many flow and

steady flame. (a) Chemiluminescence image; (b) velocity vector field and dilatationerature field; (e) temperature field; (f) pressure field; (g) and (h) close view of flamem (representative case, steady flame).

Fig. 12. Comparison of the heat release rate TF with the flame surface area TFcomputed using several definitions of the flame end point. Parameter values are:U ¼ 0:8; V ¼ 100 cm=s; d ¼ 2:0 mm and l ¼ 3:0 mm.


flame parameters are predicted qualitatively and quantitativelycorrect. This gives confidence that the time-dependent flame andflow can be also accurately computed.

5.2.2. Kinematics of excited flamesBoth real and numerical experiments on pure tone excited

flames show the periodic formation of undulations. These wavestravel along the flame front with a velocity of the order of magni-tude of the mean flow velocity. The shape and temporal evolutionof these perturbations is extensively studied and reported, and wewill not repeat this. Instead, we will concentrate on new aspects,which could be useful for future flame models.

The important question for the development of kinematic-typemodels is: suppose that a model correctly predicts the flame frontmotion, then does the flame surface area oscillation adequatelydetermine the heat release rate oscillation? Physically this isequivalent to the assumption of constant sL over the entire flamefront.

To check the validity of the kinematic approach, the deviation ofthe flame area response from the heat release rate response shouldbe studied. The presence of such a difference could indicate that ef-fects of nonconstant sL and/or heat transfer from flame to burnerdeck play a significant role.

Fig. 11 presents the time history of the simulated response ofthe representative flame to a velocity oscillation. The flame surfacearea is calculated using the flame end point definition as the heatrelease rate inflection point. The relative value of a variable isdefined as its current deviation from the mean value scaled withthis mean value. For a frequency of 200 Hz two periods of oscilla-tion were sufficient to reach a steady oscillation. The relative heatrelease oscillation q0ðtÞ and downstream velocity fluctuation u0dðtÞare in agreement with the corresponding points of the TF and TMelement Muu presented in Fig. 8.

The relative oscillation of the flame surface area A0ðtÞ has ampli-tude and phase which are close to the corresponding values of therelative heat release fluctuation for this particular excitation fre-quency. All the responses are delayed relative to the imposedvelocity oscillation as expected. Conducting similar calculationsfor a range of excitation frequencies, we can compare the thermalTF and flame area TFA. Fig. 12 presents the results of this calcula-tion. Points marked by open circles correspond to the thermal TFobtained via pure tone high amplitude (10%) excitation of theflame. The points agree well with the TF calculated via the lowamplitude excitation of the flame by a stepwise velocity jump. Thisobservation cross-validates the correctness of both TF calculationprocedures. Points marked by squares represent the flame areaTF. Both gain and phase are in good agreement with the thermal

Fig. 11. Temporal evolution of the relative upstream u0up and downstream u0dvelocity, the flame heat release rate q0 and surface area A0 subject to a perturbationwith a frequency of 200 Hz. Parameter values are: U ¼ 0:8; V ¼ 100 cm=s;d ¼ 2:0 mm and l ¼ 3:0 mm; the amplitude of excitation is 10% of the meanvelocity.

TF, which emphasizes the dominant role of the surface area oscil-lation in flame thermo-acoustic behavior.

For this example it is useful to demonstrate the effect of themethod to determine the flame end point. As mentioned above,the surface area time history, indicated by A0ðtÞ in Fig. 11 andmarked by squares in Fig. 12, is calculated via the length of theflame line from the tip to the end point, for which we take theinflection point of the heat release. Despite the large amplitudeof the velocity excitation and response, the noise level of the flamearea signal is significant. This fact hampers using a stepwise flameperturbation to calculate the flame area TF.

One of the sources of (numerical) noise of the flame area timeevolution is the difficulty to accurately determine the (motion of)the flame end point. Therefore, two other methods were tested.The dash-dotted lines in Fig. 12 indicated by A0truncated and thedotted line indicated by A0isotherm are computed using procedure(i) and (ii), respectively, explained in Section 4.1.

All flame area TFs give similar values for the time lag but differ-ent saturation frequencies. Qualitatively, the general trend of allflame area TF is similar. However, quantitatively the gain of theTFs reconstructed from A0truncated is significantly overpredicted.

5.2.3. Acoustically excited flowBoth experimental and numerical results of periodically excited

flow show the presence of structures which can be described interms of convective traveling waves. This kind of flow behaviorhas been observed earlier and is well-documented [20,24,25].However, in previous research only the downstream part, relativeto the burner outlet, of the jet was accessible for measurements,and therefore the question remains: what is the ‘starting point’of the convective wave? In this study, measurements and simula-tions of oscillating flow, both upstream and downstream of theburner deck and flame, allow us to investigate the details of thetraveling wave formation and may support the development of anew theory for thermo-acoustic behavior of flames.

Both real and numerical experiments show the same flow struc-ture. However, numerical results are more detailed and we willonly consider these. Due to flow contraction/expansion the meanand oscillating components of the axial velocity vary significantlyalong the streamlines. Therefore, a more convenient indicator totrace the oscillating flow structure is the ratio of the oscillating

Fig. 14. Temporal evolution of the relative streamwise velocity component velocityat the slit center point line, point C; transversal velocity component in the nearupstream flow line, point U; transversal component in the near downstreamlocation line; point D. Parameter values are: U ¼ 0:8; V ¼ 100 cm=s; d ¼ 2:0 mm;

l ¼ 3:0 mm and f ¼ 200 Hz.


component and the mean value of the axial velocity. The meanvelocity is calculated as the time average (over one period) of thevelocity, at each point in space. The oscillating component is thedifference between the (local) mean and instant velocity.

Fig. 13 presents the distribution of the relative flow oscillationalong the centerline of the burner slit. Different lines correspondto progressing phases of the imposed oscillation. Far upstream ofthe burner plate the perturbation is uniform and in phase. Thisplane acoustic wave feature of the flow persists to the vicinity closeupstream of the burner plate. Waves of convective nature areformed in the immediate proximity of the cross-section of the bur-ner deck. The waves convectively travel to the flame front wherethey disappear and once again a plane wave, with a new amplitudeand phase, is recovered.

The large jump in the relative oscillations at the flame tip is anartefact of the data processing procedure. In Fig. 13 this part is cov-ered with the gray rectangle. The reason for this is that the mea-surement points in this zone are alternatingly in the burnt andunburnt gas mixture, which leads to an artificially large relativeamplitude of the velocity oscillation.

As indicated above, the flame–flow interaction in the flame footzone could hypothetically determine the TF offset characteristicsand the flame retroaction to the upstream flow. Because of this itis interesting to have a look at the oscillating flow dynamics inthe burner plate vicinity. Fig. 14 presents the time evolution ofthe oscillating part of the flow velocities computed at three pointsnear the slit. All velocities are scaled with the mean flow velocity atthe inflow. The line marked as V s

C represents the axial oscillatingvelocity in the slit center point (C in Fig. 14). The lines indicatedas V t

U and V tD are the transversal component of the oscillating flow

velocity at 0.5 mm upstream (point U) and downstream (point D)of the burner deck, respectively; see the right part of Fig. 14. Theoscillating vertical velocity component on the centerline is in anti-phase with the transversal upstream velocity, which is intuitivelyclear – the moment when the vertical velocity in the slit reachesthe largest upward value coincides with the maximum of the in-ward directed value of the upstream transversal velocity. At thesame time, the flow dynamics in the near downstream part ofthe flow is more complicated. The phase of oscillation of the hori-zontal velocity in point D is some value ahead of the oscillation inthe slit center point.

6. Discussion

The good agreement between experimental and numericalresults for the TF, flow fields, stationary and excited flames ob-tained for a wide range of parameters, convinces us that both theexperimental and numerical approaches are adequate to describe

Fig. 13. Distribution of relative streamwise flow velocity along the slit centerline.Different lines correspond to four different phases of the applied perturbation, withfrequency 200 Hz and relative amplitude 0.1. Parameter values are: U ¼ 0:8; V ¼100 cm=s; d ¼ 2:0 mm and l ¼ 3:0 mm.

the essential phenomena in thermo-acoustic flame behavior ofmulti-slit Bunsen flames.

Regarding the measurement procedure the following questionshave been clarified. First, is chemiluminescence of the flame adirect and appropriate indicator of the heat release rate for thepractically relevant case of small flames? This question should beanswered affirmative because in our model there are no assump-tions concerning the relation between chemiluminescence andheat release rate. The second controversial aspect of the TFmeasurement approach is: were is the ‘input’ of the burner–flamesystem, or more precisely, where should the flow velocity pertur-bation be measured? Results above show that for the testedflame–burner configuration, the appropriate location to measurethe velocity perturbation is not too close to the burner deck. Thisis valid, if the aim of the flame characterization is to relate the heatrelease rate to the plane wave acoustic perturbation. For our casethis location should be more than a few millimeters upstream ofthe burner plate. In the close vicinity of the flame, a convectiveflow perturbation is generated and the plane acoustic wave behav-ior is affected. This result is in contradiction with [3] where therecommended position to measure the TF is as close as possibleto the flame cone. As follows from the spatially resolved study ofthe excited flow (see Fig. 13) the flow inside the slit and closeup/downstream is dominated by convective waves. If the flameTF is measured inside the burner deck or close above the slit outletthen the relation between the local velocity and acoustic velocityin a plane wave in the burner upstream part should be incorpo-rated. In [13,30] such a strategy was successfully applied, howeverthe question remains whether it is justified to use cold flow theoryto estimate the acoustic features of a flame.

Regarding the numerical simulations, the agreement betweenexperimental and simulation results confirms that the underlyingmodel, assuming incompressible (low-Mach number) reacting flowwith one-step chemistry, is sufficient to reproduce the global flamethermo-acoustic properties as well as the spatially and temporallyresolved flame structure and flow. The fact that the measuredshape and height of the flames are correctly reproduced by ourmodel is direct evidence of the adequacy of the chemical kineticsused. The model captures the thermo-acoustic TF of the multipleflame configuration. Furthermore, parameters for stationary andexcited flames are also well reproduced. Accordingly, we concludethat this model includes all important physical phenomena, andtherefore provides a convenient tool to investigate flame–flowinteraction. Currently there are no arguments to include detailedchemistry and/or sophisticated diffusion models. Furthermore,the computation of the TF does not require the inclusion of com-pressibility effects. However, this might not be true for the TM.


This question was also discussed in [3] and requires furtherinvestigation.

An analytical model of the interaction between premixed Bun-sen-type flames and flow oscillations requires the identificationof those phenomena which are responsible for the flame response.The key question is: what is the level of complexity required toaccurately predict the thermo-acoustic response of the flame?

Nowadays, kinematic models to describe the premixed flame–acoustics interaction are widely studied. Are these modelsadequate to predict the thermal TF of practically interesting config-urations where the perforations, and accordingly also the flameheights, are of the order of 1 mm? This question can be split intwo parts. The first one is: how to model properly the flame sheetkinematics? The second question is: how do we relate the heat re-lease rate fluctuations to the movement of the flame sheet?

With regard to the first question we note the following. Allexperimental and numerical results of the TF show that the timedelay is proportional to the convective time s ¼ H=V . This time de-lay might vary due to variations in flow velocity (Fig. 4), slit width(Fig. 5), pattern pitch (Fig. 6) or mixture equivalence ratio (Fig. 7).Nowadays, it is hypothesized that the time lag should be explainedin terms of convective structures of the upstream flow. The obser-vation that convective waves also occur in our flame configurationand that they originate close to the burner deck (see Fig. 13) doesnot contradict this hypothesis. Accordingly, in order to predict thecorrect time delay, analytical models should include convectivewave features.

The occurence of maxima and minima in the TF gain as well asthe saturation behavior of the TF gain and phase can be attributedto a weakly frequency dependent offset component of the TF. Onceagain, the smaller the flame the more pronounced the TF offset is.The saturation frequency is lower for the flames with smallervelocity (Fig. 4), smaller slit width (Fig. 5), smaller pitch (Fig. 6)or richer mixture composition (Fig. 7). This observation is in agree-ment with the earlier mentioned hypothesis that the TF offset isrelated to the flame foot dynamics and probably resembles themechanism describing the interaction between a surface burner-stabilized flame and acoustics waves. Accordingly, to predict theTF offset feature, analytical models should incorporate flame footbehavior.

The available data are not conclusive of the cause of the TF gainovershoot. Two hypotheses, which require further investigation,can be proposed. First, the overshoot is due to the low-frequencyfeatures of the offset component in the TF. The other possible rea-son is the interaction of adjacent flames. The flame-to-flame influ-ence should be included anyhow in an analytical model in order topredict the influence of the pitch on the TF (see Fig. 6).

The second aspect is how to relate the flame surface area fluc-tuations to the heat release rate oscillations. As is apparent fromFig. 12 the TFs for flame area and heat release rate correlate well.It means that the problem of variation of the normal flame propa-gation speed over the flame surface due to heat transfer to the bur-ner deck or flame stretch (because of flow straining and flamecurvature) is not very pronounced, even for small flames. Instead,another problem arises: how to specify the flame end point? Forexample, the physically plausible way to end the flame surface inthe point of maximal decay of the heat release (inflection point)leads to a good prediction of the flame TF. However, this choice re-quires an accurate numerical simulation of the flame foot zone. Theartificial truncation of the flame front by the slit edges can result ina significant error in the computed TF (see Fig. 12). Taking an iso-therm as flame front, e.g. the 1200 K isotherm, hardly can be ac-cepted, since it would imply that the flame front is defined evenin points where there is no heat release at all; see Fig. 10c. Our re-sults reveal these problems, show possible causes of discrepanciesbut give no answers how to eliminate these.

The difficulty to identify the flame TF in a consistent way andsubsequently use it to calculate the TM was recognized earlierfor compressible flow simulation of a flame excited by the puretone with a single frequency [3]. It was surmised that, because ofreflections of the acoustic wave in the downstream section of theburner, the point to measure the excitation velocity should be asclose as possible to the flame, in order to obtain a Helmholtz num-ber He less than 10�2. It is defined as He :¼ L=k, where L is the dis-tance between a mean flame position and the measurement pointand k the wavelength of the acoustic perturbation. In the incom-pressible flow approximation used here the speed of sound, andaccordingly the acoustic wavelength, are assumed to be infinitelylarge leading to He = 0.

Another aspect of the problem to link the flame TF and TM forsmall flames is related to the influence of the heat transfer betweenthe flame and burner deck, which should be considered as well,just as for flat burner surface stabilized flames [6,39]. However,for larger flames, when the role of the flame foot zone is lessimportant the relation between TF and TM can be derived fromthe Rankine–Hugoniot jump conditions. Discussion of the link be-tween the TF and TM can be found in [3], details of the derivationand further references are presented in [5]. The resulting formulashows the linear proportionality of the TF and Muu, i.e.,

Muu ¼Tburnt

Tunburnt� 1

� �TFþ 1: ð8Þ

For the flames studied here the comparison of the directly mea-sured (simulated) element Muu of the TM (see solid lines in Fig. 8cand d) with the one calculated from the corresponding TF accord-ing to (8) (dotted lines in the same figures) shows perfect correla-tions even for the smallest tested flame size/flow velocity. This fact,on the one hand, once again confirms the consistency of the resultsobtained, and, on the other hand, supports the conclusion that theflame to burner deck thermal interaction effects are not too sub-stantial for the flames studied here. Certainly, this conclusionshould not be extended to any small flames. Admittedly, the linkbetween TF and TM for the case of small Bunsen flames is some‘superposition’ of the Rankine–Hugoniot and the relation in [6].

If the hypothesis about the creation of traveling waves via thejet contraction–expansion modulation is valid, then at least a peri-odic in–out motion of gas along the burner plate should be ob-served. Fig. 14 shows this type of motion near the flame foot andjust beneath the slit. On the basis of the available data it is difficultto assess whether this effect is due to the acoustic jet or simply thepresence of the flame. Clarification of this question requires a spe-cial numerical experiment with a carefully tuned model.

7. Conclusions

The objective of our experimental and numerical investigationof the thermo-acoustic behavior of multiple-slit Bunsen flames isto assess the applicability of the thermal transfer function conceptas well as to validate the experimental and simulation tools used.

The investigation shows that our (numerical) model is capableto reproduce all aspects observed in the experiments. A parametricstudy is carried out, where the mean flow velocity V, the equiva-lence ratio U, the slit width d and the distance between slits l arevaried. It is shown that the measured and computed flame transferfunction are qualitatively the same. Furthermore, good qualitativeagreement between measured and computed parameters isachieved.

For the flames studied here, relation (8), which is based on theRankine–Hugoniot jump conditions, proved to accurately repro-duce the relation between the thermo-acoustic TF and the elementMuu of the acoustic TM.


The spatially resolved steady flame and the temporally resolvedacoustically excited flame indicate that all essential effects offlame–acoustics interaction are captured by our model of incom-pressible flow, with one-step chemical kinetics and a simple diffu-sion model. The comparison also gives additional confidence in themeasurement techniques (i.e. chemiluminescence and heated wiretechniques) used during the experiments.

Synergy of numerical simulation and experiments allows us toobtain insight into the physics of flame–acoustics interactionand, subsequently to formulate recommendations for analyticalmodels. In particular, we recommend the following. First, to pre-dict the time lag it is necessary to combine the kinematic modelwith a model for the creation of convective waves at the burneroutlet. Second, to predict the TF saturation, the jagged behaviorand, probably the effect of the TF gain overshoot, it is necessaryto develop a method to describe the dynamics of the flame anchor-ing zone. Finally, inclusion of a model for the flame retroaction tothe upstream flow and flame-to-flame interaction in the multipleflame configurations could improve the accuracy and capabilityof the kinematic model to predict the flame TF of practically inter-esting burners.

An additional result of our investigation is the clarification ofthe difficulties encountered in the description of flame thermo-acoustics interaction. One of the difficulties is how to relate theflame surface area oscillation to the heat release rate oscillation.This is partially due to the ambiguity of the flame anchoring pointdefinition. Explanation of these issues requires an additionalinvestigation.

Acknowledgment

This research is supported by the European Marie CurieResearch Training Network (project: ‘‘AETHER”).

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Experimental and numerical investigation of the acoustic response of multi-slit Bunsen burners

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